Risk-Neutral Probabilities

[Pages:12]Debt Instruments and Markets

Professor Carpenter

Risk-Neutral Probabilities

Concepts

Risk-neutral probabilities Risk-neutral pricing Expected returns True probabilities

Reading

Veronesi, Chapter 9 Tuckman, Chapter 9

Risk-Neutral Probabilities

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Debt Instruments and Markets

Professor Carpenter

No Arbitrage Derivative Pricing

Last lecture, we priced a derivative by constructing a replicating portfolio from the underlying zeroes:

?We started with a derivative with a payoff at time 0.5. The payoff depended on the time 0.5 price of the zero maturing at time 1.

?We modeled the random future price of the zero and the future payoff of the derivative.

?We constructed a portfolio of 0.5-year and 1year zeroes with the same payoff of the derivative by solving simultaneous equations.

?We then set the price of the derivative equal to the value of the replicating portfolio.

General Bond Derivative

?Any security whose time 0.5 payoff is a function of the time 0.5 price of the zero maturing at time 1 can be priced by no arbitrage.

?Suppose its payoff is Ku in the up state, and Kd in the down state:

Time 0

0.5-year zero 1-year zero

0.973047 0.947649

General portfolio

0.973047 N0.5 + 0.947649 N1

General derivative ?

Time 0.5 1 0.972290 1N0.5 + 0.97229N1

K u

1 0.976086

1N0.5 + 0.976086N1

K d

Risk-Neutral Probabilities

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Debt Instruments and Markets

Professor Carpenter

Replicating and Pricing the General Derivative

1) Determine the replicating portfolio by solving the equations 1N0.5 + 0.97229N1 = Ku

1N0.5 + 0.96086N1 = Kd for the unknown N's. (The two possible K's are known.)

2) Price the replicating portfolio as 0.973047N0.5 + 0.947649N1

This is the no arbitrage price of the derivative.

Risk-Neutral Probabilities

?Finance: The no arbitrage price of the derivative is its replication cost.

?We know that's some function of the prices and payoffs of the basic underlying assets.

?Math: We can use a mathematical device, risk-neutral probabilities, to compute that replication cost more directly.

?That's useful when we only need to know the price of the replicating portfolio, but not the holdings.

Risk-Neutral Probabilities

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Debt Instruments and Markets

Professor Carpenter

Start with the Prices and Payoffs of the Underlying Assets

?In our example, the derivative payoffs were functions of the time 0.5 price the zero maturing at time 1.

?So the underlying asset is the zero maturing at time 1 and the riskless asset is the zero maturing at time 0.5.

?The prices and payoffs are, in general terms:

Time 0

d0.5 d1

Time 0.5 1

du

0.5 1

1 dd

0.5 1

Find the "Probabilities" that "Risk-Neutrally" Price the Underlying Risky Asset

?Find the "probabilities" of the up and down states, p and 1-p, that make the price of the underlying asset equal to its "expected" future payoff, discounted back at the riskless rate.

?I.e., find the p that solves "Risk-Neutral Pricing Equation" (RNPE)

Price = discounted "expected" future payoff

for the underlying risky asset.

?In our example, this is the zero maturing at time 1, so

d1 = d0.5[ p ?0.5 d1u + (1 - p) ?0.5 d1d ]

p=

d1 d0.5

-0.5

d1d

du

0.5 1

-0.5

d1d

Risk-Neutral Probabilities

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Debt Instruments and Markets

Professor Carpenter

Example of p

In our example,

0.947649 - 0.976086

p = 0.973047

= 0.576

0.972290 - 0.976086

1- p = 0.424

Result

?The same p prices all the derivatives of the underlying risk-neutrally. ?I.e., if a derivative has payoffs Ku in the up state and Kd in the down state, its replication cost turns out to be equal to

Derivative price = d0.5[ p ? Ku + (1- p) ? Kd ]

using the same p that made this risk-neutral pricing equation (RNPE) hold for the underlying asset. ?That is, a single p makes the RNPE

price = discounted "expected" future payoff

hold for the underlying and all its derivatives.

Risk-Neutral Probabilities

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Debt Instruments and Markets

Professor Carpenter

Examples of Risk-Neutral Pricing

With the risk-neutral probabilities, the price of an asset is its expected payoff multiplied by the riskless zero price, i.e., discounted at the riskless rate:

call option: (0.576 ? 0 + 0.424 ?1.086) ? 0.9730 = 0.448

or, 0.576 ? 0 + 0.424 ?1.086 = 0.448 1.0277

Class Problem:

Price using

the the

put option with payoffs Ku=2.71 risk-neutral probabilities.

and

Kd=0

Examples of Risk-Neutral Pricing...

1-year zero: (0.576 ? 0.9723 + 0.424 ? 0.9761) ? 0.9730 = 0.9476

or, 0.576 ? 0.9723 + 0.424 ? 0.9761 = 0.9476 1.0277

0.5-year zero (riskless asset):

(0.576 ?1+ 0.424 ?1) ? 0.9730 = 0.9730

or, 0.576 ?1+ 0.424 ?1 = 0.9730 1.0277

Risk-Neutral Probabilities

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Debt Instruments and Markets

Professor Carpenter

Why does the p that makes the RNP Equation hold for the underlying also make the RNPE work for all its derivatives?

1)By construction, p makes price of underlying risky asset = discount factor x [p x underlying's up payoff + (1-p) x underlying's down payoff].

2)It's also always true for any p that price of riskless asset = discount factor = discount factor x [p x 1 + (1-p) x 1]

so the p that works for the underlying also works for the riskless asset, because any p does.

3)Therefore, this p also works for any portfolio of these two assets. I.e., for any portfolio with holdings N0.5 and N1: N0.5 x price of riskless asset + N1 x price of underlying = disc.factor x [p x (N0.5 x 1 + N1 x underlying's up payoff ) + (1-p) x (N0.5 x 1 + N1 x underlying's down payoff ) ]

i.e., portfolio price = disc.factor x [p x portfolio's up payoff + (1-p) x portfolio's down payoff]

Since every derivative of the underlying is one of these portfolios, the RNPE, using the same p, holds for all of them too.

Class Problem

?Suppose the time 0 price of the zero maturing at time 1 is slightly lower:

Time 0

0.973047 0.947007

Time 0.5 1 0.972290

1 0.976086

?What would be the risk-neutral probabilities p and 1-p of the up and down states?

Risk-Neutral Probabilities

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Debt Instruments and Markets

Professor Carpenter

Class Problem

1)Price the call using these new RN probs:

Time 0

Time 0.5 1

0.973047 0.947007

0.972290 0

Call?

1

0.976086

1.086

2)Calculate the replication cost of the call the old way and verify that it matches the price above. Hint: the payoffs are unchanged so the replicating portfolio is still N0.5 = -278.163 and N1 = 286.091.

Expected Returns with RN Probs

?Note that we can rearrange the risk-neutral pricing equation, price = discounted "expected" payoff, as

V = d0.5[ p ? Ku + (1- p) ? Kd ], or

V = p ? Ku + (1- p) ? Kd 1+ r0.5 /2

p ? Ku

+ (1- V

p) ? Kd

= 1+ r0.5 /2

?I.e., "expected" return = the riskless rate.

(Here return is un-annualized. )

?Thus, with the risk-neutral probabilities, all assets have the same expected return--equal to the riskless rate.

?This is why we call them "risk-neutral" probabilities.

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